Solve f(a+1/a)+g(a-1/a+1)=3

$$f ( \frac { a + 1 } { a } ) + g ( \frac { a - 1 } { a + 1 } ) = 3$$

Solve for f

$$f=-\frac{a\left(ag-3a-g-3\right)}{\left(a+1\right)^{2}}$$

$$a\neq 0\text{ and }a\neq -1$$

Steps for Solving Linear Equation

Multiply both sides of the equation by a\left(a+1\right), the least common multiple of a,a+1.

$$f\left(a+1\right)\left(a+1\right)+ga\left(a-1\right)=3a\left(a+1\right)$$

Multiply a+1 and a+1 to get \left(a+1\right)^{2}.

$$f\left(a+1\right)^{2}+ga\left(a-1\right)=3a\left(a+1\right)$$

Use binomial theorem \left(p+q\right)^{2}=p^{2}+2pq+q^{2} to expand \left(a+1\right)^{2}.

$$f\left(a^{2}+2a+1\right)+ga\left(a-1\right)=3a\left(a+1\right)$$

Use the distributive property to multiply f by a^{2}+2a+1.

$$fa^{2}+2fa+f+ga\left(a-1\right)=3a\left(a+1\right)$$

Use the distributive property to multiply ga by a-1.

$$fa^{2}+2fa+f+ga^{2}-ga=3a\left(a+1\right)$$

Use the distributive property to multiply 3a by a+1.

$$fa^{2}+2fa+f+ga^{2}-ga=3a^{2}+3a$$

Subtract ga^{2} from both sides.

$$fa^{2}+2fa+f-ga=3a^{2}+3a-ga^{2}$$

Add ga to both sides.

$$fa^{2}+2fa+f=3a^{2}+3a-ga^{2}+ga$$

Reorder the terms.

$$2af+fa^{2}+f=3a^{2}+ag+3a-ga^{2}$$

Combine all terms containing f.

$$\left(2a+a^{2}+1\right)f=3a^{2}+ag+3a-ga^{2}$$

The equation is in standard form.

$$\left(a^{2}+2a+1\right)f=3a^{2}+ag+3a-ga^{2}$$

Divide both sides by a^{2}+2a+1.

$$\frac{\left(a^{2}+2a+1\right)f}{a^{2}+2a+1}=\frac{a\left(3+g+3a-ag\right)}{a^{2}+2a+1}$$

Dividing by a^{2}+2a+1 undoes the multiplication by a^{2}+2a+1.

$$f=\frac{a\left(3+g+3a-ag\right)}{a^{2}+2a+1}$$

Divide a\left(3a+g+3-ga\right) by a^{2}+2a+1.

$$f=\frac{a\left(3+g+3a-ag\right)}{\left(a+1\right)^{2}}$$

Solve for a (complex solution)

$$\left\{\begin{matrix}a=\frac{-\sqrt{9+6g+g^{2}-8fg}+g-2f+3}{2\left(f+g-3\right)}\text{, }&\left(g\neq 1\text{ or }f\neq 2\right)\text{ and }\left(arg(g+3)\geq \pi \text{ or }f\neq 0\right)\text{ and }\left(g\neq -3\text{ or }f\neq 0\right)\text{ and }f\neq 3-g\\a=\frac{\sqrt{9+6g+g^{2}-8fg}+g-2f+3}{2\left(f+g-3\right)}\text{, }&\left(g\neq 1\text{ or }f\neq 2\right)\text{ and }\left(f\neq 0\text{ or }arg(g+3)<\pi \text{ or }g\neq 1\right)\text{ and }\left(arg(g+3)<\pi \text{ or }f\neq 0\right)\text{ and }\left(g\neq -3\text{ or }f\neq 0\right)\text{ and }f\neq 3-g\text{ and }g\neq 0\\a=-\frac{3-g}{3\left(1-g\right)}\text{, }&g\neq 3\text{ and }g\neq 1\text{ and }g\neq 0\text{ and }f=3-g\end{matrix}\right.$$